) cross sections of y, nb, and ta

13
PHYSI CA L RE VIEW C VOLUME 4, NUMBER 2 AUGUST 1971 Equilibrium and Nonequilibrium Contributions to the (p, n) Cross Sections of Y, Nb, and Tag S, M. Grimes, J. D. Anderson, B. A. Pohl, J. W. McClure, and C. Wong Lawrence Radiation Laboratory, Livermore, Calzfornia 94550 (Received 19 April 1971) The neutron spectra produced by the Y(p, n) ~Zr, Nb(P, n)~3Mo, and Ta(p, n) W reac- tions have been measured at five angles between 15 and 135 for proton energies between 7.8 and 14. 8 MeV. It is concluded that the data at low bombarding energies are consistent with a compound-nuclear-reaction mechanis m. A constant-temperature level density for energies below the neutron binding energy is found to be appropriate for the residual nuclei 9Zr, ~ Mo, and W, with nuclear temperatures 0=0.70, 0. 68, and 0.54 MeV, respectively. For bombarding energies above 10 MeV contributions from noncompound processes appear and at 14. 8 MeV these are comparable in magnitude to those produced by compound-nuclear (p, n) reactions. On the basis of the angular distributions and the bombarding-energy depen- dence the relative fraction of the noncompound cross section due to direct and to pre-equilib- rium reaction mechanisms is estimated. The extracted average doorway-state width varies from 350 keV near A. =90 to about 1 MeV for A- 180. I. INTRODUCTION A recent study' of the ~'V(P, n)"Cr and ' Co(P, n)- "Ni reactions has yielded estimates of 160 keV for the doorway-state widths. Careful examina- tion of the dependence of the neutron spectra on bombarding energy and residual excitation made it possible to divide the neutron spectra into equi- librium and nonequilibrium portions. Comparison of this latter component with theoretical predic- tions' ' enabled a further separation into direct and pre-equilibrium contributions to be made. It was suggested that nuclei with lower nuclear tem- peratures might yield better information on the pre-equilibrium reaction mechanism, because the equilibrium contribution to a given region of the spectrum would decrease more rapidly with in- creasing bombarding energy, making the pre-equi- librium contribution relatively more prominent. Measurements of the neutron spectra produced in the 8 Y(p, n)'~Zr, 9'Nb(p, rp Mo, and '8'Ta(p, n}- '"W reactions were carried out both to test this prediction and to investigate the dependence on A of the pre-equilibrium state width. The statistical theory' predicts that the differ- ential cross section for emission of particles of energy E integrated over angle can be related to the level density p(U) of the residual nucleus as follows: o(e) ~ Eo( )pe(U)' where e is the channel kinetic energy and o, (e} is the capture cross section for the inverse reaction at an energy c. Although many experiments have yielded results consistent with the statistical the- ory, other measurements" have indicated that the continuum spectra cannot be described entirely by Eil. (1). Specifically, the level-density parame- ters obtained from such measurements were func- tions of the bombarding energy as well as the re- sidual excitation. The logical explanation" for this behavior is that nonequilibrium processes al- so contributed to the continuum spectra. A specif- ic model' ' for such a mechanism has been pro- posed and both the bombarding energy and resid- ual excitation dependence were calculated as func- tions of the single-particle level density. Thus, experimental information on the above energy de- pendences of pre-equilibrium contributions pro- vides a test of the assumptions of the current model. II. THEORY oa 8 + abs ~ 8 con /~ints (2} where O, b, is the absorption cross section for particle a, I'& „„ is the width of the intermediate The formal treatment of intermediate structure by Feshbach, Kerman, and Lemmer' contains the basic concepts required to analyze pre-equilibri- um emission, i.e. , the continuum decay of "door- way" states. In this basic treatment the cross sections are calculated from the resonance param- eters of the intermediate states rather than the various level densities themselves. The extention to the level-density case is carried out as in the statistical compound-nucleus case. Feshbach, Kerman, and Lemmer' have shown that the pre-equilibrium cross section for the pro- cess (a, P) is 607

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Page 1: ) Cross Sections of Y, Nb, and Ta

PHYSI CA L RE VIEW C VOLUME 4, NUMBER 2 AUGUST 1971

Equilibrium and Nonequilibrium Contributions to the (p, n) Cross Sections

of Y, Nb, and Tag

S, M. Grimes, J.D. Anderson, B.A. Pohl, J.W. McClure, and C. WongLawrence Radiation Laboratory, Livermore, Calzfornia 94550

(Received 19 April 1971)

The neutron spectra produced by the Y(p,n) ~Zr, Nb(P, n)~3Mo, and Ta(p, n) W reac-tions have been measured at five angles between 15 and 135 for proton energies between 7.8and 14.8 MeV. It is concluded that the data at low bombarding energies are consistent with acompound-nuclear-reaction mechanis m. A constant-temperature level density for energiesbelow the neutron binding energy is found to be appropriate for the residual nuclei 9Zr, ~ Mo,and W, with nuclear temperatures 0=0.70, 0.68, and 0.54 MeV, respectively.

For bombarding energies above 10 MeV contributions from noncompound processes appearand at 14.8 MeV these are comparable in magnitude to those produced by compound-nuclear(p,n) reactions. On the basis of the angular distributions and the bombarding-energy depen-dence the relative fraction of the noncompound cross section due to direct and to pre-equilib-rium reaction mechanisms is estimated.

The extracted average doorway-state width varies from 350 keV near A. =90 to about 1 MeVfor A- 180.

I. INTRODUCTION

A recent study' of the ~'V(P, n)"Cr and ' Co(P, n)-"Ni reactions has yielded estimates of 160 keVfor the doorway-state widths. Careful examina-tion of the dependence of the neutron spectra onbombarding energy and residual excitation madeit possible to divide the neutron spectra into equi-librium and nonequilibrium portions. Comparisonof this latter component with theoretical predic-tions' ' enabled a further separation into directand pre-equilibrium contributions to be made. Itwas suggested that nuclei with lower nuclear tem-peratures might yield better information on thepre-equilibrium reaction mechanism, because theequilibrium contribution to a given region of thespectrum would decrease more rapidly with in-creasing bombarding energy, making the pre-equi-librium contribution relatively more prominent.

Measurements of the neutron spectra producedin the 8 Y(p, n)'~Zr, 9'Nb(p, rp Mo, and '8'Ta(p, n}-'"W reactions were carried out both to test thisprediction and to investigate the dependence on Aof the pre-equilibrium state width.

The statistical theory' predicts that the differ-ential cross section for emission of particles ofenergy E integrated over angle can be related tothe level density p(U) of the residual nucleus asfollows:

o(e) ~ Eo( )pe(U)'

where e is the channel kinetic energy and o,(e} isthe capture cross section for the inverse reactionat an energy c. Although many experiments haveyielded results consistent with the statistical the-

ory, other measurements" have indicated that thecontinuum spectra cannot be described entirely byEil. (1). Specifically, the level-density parame-ters obtained from such measurements were func-tions of the bombarding energy as well as the re-sidual excitation. The logical explanation" forthis behavior is that nonequilibrium processes al-so contributed to the continuum spectra. A specif-ic model' ' for such a mechanism has been pro-posed and both the bombarding energy and resid-ual excitation dependence were calculated as func-tions of the single-particle level density. Thus,experimental information on the above energy de-pendences of pre-equilibrium contributions pro-vides a test of the assumptions of the currentmodel.

II. THEORY

oa 8 + abs ~ 8 con /~ints (2}

where O,b, is the absorption cross section forparticle a, I'& „„is the width of the intermediate

The formal treatment of intermediate structureby Feshbach, Kerman, and Lemmer' contains thebasic concepts required to analyze pre-equilibri-um emission, i.e., the continuum decay of "door-way" states. In this basic treatment the crosssections are calculated from the resonance param-eters of the intermediate states rather than thevarious level densities themselves. The extentionto the level-density case is carried out as in thestatistical compound-nucleus case.

Feshbach, Kerman, and Lemmer' have shownthat the pre-equilibrium cross section for the pro-cess (a, P) is

607

Page 2: ) Cross Sections of Y, Nb, and Ta

608 GRIMES, ANDERSON, POHL, McCLURE AND WONG

state for emission of particle P, and I';„, is thetotal width of the intermediate state. The widthI ~ for emission of particle P leaving the nucleusin a particular state can be shown to be~

(3)

where D is the average spacing between states and

T, is the transmission coefficient for angular mo-mentum l. If the inverse capture cross section isintroduced in place of the sum in E11. (3) and if thewidth to all residual states is summed, this ex-pression becomes'

I S„„dE=, ',"'Ep'(E, -E)dE,(2S+ 1)mes;„,P nt~*

where m and S are the mass and spin of the emit-ted particle, respectively, p'(U) is the density ofstates in the residual nucleus which can be reachedby emission from an intermediate state, and

p;„,(E ) is the density of intermediate states in thecompound nucleus at an excitation energy F.*. IfE11. (4) is substituted into E11. (2), the followingresult is obtained:

( )(2S+ I)m G,„,(E,)(ra;„gE)Ep'(E, —E)

(5)

intermediate widths requires an assumption as tothe form of p;„,(E~), while p'(U) can be determinedfrom the shape of the emission spectrum. Theform of Eq. (5) is completely analogous to thecompound-nucleus case where the relevant leveldensities and widths refer to the particle-hole orcompound states, respectively.

III. EXPERIMENTAL PROCEDURE

Neutron spectra. from the 'Y(P, n)' Zr, "Nb-

(P, n)"Mo, and "'Ta(p, n)'"W reactions for anglesat 30= intervals from 15 to 135"were obtained withthe experimental technique described previously. "The Livermore 2.3-m variable energy cyclotronproduced protons with energies from 7.8 to 14.8MeV. Neutrons resulting from the bombardmentof self-supporting targets of Y, Nb, and Ta weredetected in NE213 scintillators with the neutronenergy determined by the time of flight over a10.8-m flight path. A linear bias eliminatedpulses produced by recoil protons with energiesless than 1.6 MeV and y-produced background wasreduced by utilizing pulse-shape discrimination.A typical (P, n) spectrum is shown in Fig. 1.

TABLE I. Nuclear temperatures for Zr, ~ Mo, and18i~

E1luation (5) is a model-independent expressionand does not depend on the nature of the interme-diates states. In practice, use of Eq. (5) to evaluate

Ep(MeV)

U range(Me V) 15' 45 75

Y(p, n) Zr

105 135

10'

OP

IPL

E—lo'—EO

b Ip

Ep=14.8 MeY1 1 I 1 1 1 1 & 1

8L= l35

~ ~

~ ~~ ~

10.012.3

14.8

8.710.012.3

14.0

2—42—44 —6.52—44—6.56.5—9

3.5—5.53.5—63.5-66-83.5—66-8

0.74 0.68 0.691.04 0.98 0.880.81 0.71 0.742.04 1.98 1.721.29 1.30 1.220.79 0.74 0.72

~~Nb(p, n) &3Mo

0.68 0.64 0.670.73 0.70 0.690.88 0.86 0.900.83 0.75 0.741.03 1.17 1.0 91.18 1.18 1.08

isiTa(p, n) isis

0.65 0.680.93 0.840.69 0.701.58 1.341.12 0.970.75 0.72

0.69 0.670.68 0.690.87 0.830.78 0.730.96 0.941.04 0.96

~ ~

IO~

0 I 2 3 4 5 6 7 8 910 II I2

E„(MeV) c.m.

FIG. 1. The differential cross section for the Y(p, n)-BZr reaction as a function of center-of-mass neutron en-

ergy. The bombarding energy was 14.8 MeV and the lab-oratory angle was 135'. Indicated errors are statisticalonly.

8.08.79.6

10.0

10.4

11.712.314.8

2—52 —52—44—6.72—44—6.72—44—6.72-6.72—6.72—6.7

0.49 0.55 0.520.55 0.61 0.520.62 0.53 0.600.60 0.58 0.500.86 0.75 0.680.62 0.59 0.570.91 0.81 0.800.69 0.63 0.621.12 1.18 1.111.25 1.37 1.411.71 1.42 1.35

0.53 0.540.54 0.570.59 0.510.54 0.570.62 0.600.56 0.530.77 0.740.62 0.641.03 1.051.20 1.181.31 1.26

Page 3: ) Cross Sections of Y, Nb, and Ta

EQUILIBRIUM AND NONEQUILIBRIUM CONTRIBUTIONS. . . 609

IV. DATA REDUCTION AND ANALYSIS

A. Level Densities and Pre-Equilibrium EmissionsBelow the Neutron Binding Energy

The technique of analysis was identical to thatdescribed in a previous paper. ' The continuum

portion of the (p, n) spectra was analyzed by cal-culating the "channel" cross section:

v(8, e) = a(8, E„),4+1

where A is the mass number of the residual nu-cleus, e =[(A+1)E„/A], is the channel kineticenergy, and o(8, E„), is the center-of-masscross section for the reaction. If the capture

cross section o, (e) is assumed to be constant, the

o(8, e)/e will be proportional to the level densityof the residual nucleus. In general, if pre-equi-librium or direct contributions are also present,additional terms' of the form U" will also be need-ed to fit the spectrum. Because the coefficientsof the various terms are expected to have differ-ent dependences on bombarding energy, the shapeof the continuum spectrum will change as a func-tion of bombarding energy, unless only one reac-tion mechanism is involved.

Such changes can be observed in Table I and inFigs. 2-4, which show the spectra for the "Y(P, n),

10

E =12.5 MeVP

IO W

E =l4.8 MeV

-I10

x+x ~

X ~X

X

IO

IO

xX

OTROP I CX ~

XgX~

„xx"i SYMMETRICxxx ~

X +~X X

x &x x ~ y

E =12.5 MeV

-2IO

10

x" SYMMETRIC

x ~x

XeX

+x xxy ex ~

~$ ~x

IO

IQ

LLI

IO

CLIO

b

IO

xx

X

X

x" & ~X

~ u

~ ~

XX ~

X ~X+

SOT ROP IC

E =8.7 MeV

x ~ ~xx ~

xx x ~

x SYMMETRICx x ~

5 ~x xX

IO

le ".x x

-5IO

LLI

Io '-I—

x

LLI10 — ~

x

F =10.0 MeV

~ ISOTROPICx

Xx+

X ~xxx

~ ~~ ~

x

SYMMETRIC

IO

IO

~ X0 ~xi x

x

ISOTROPIC~ x

b-8

10

-910

x xx ~

x

X

xg$x4

F =8.7 MeV

IO-IO

x ~ 3exx ISOTROPIC

IQ-I 0 I 2 5 4 5 6 7 8 9

0 (Mev)

-I I~

10

—l2

xx

FIG. 2. Neutron spectra from the Y(p, n) Zr reac-tion at three bombarding energies. The points labeled xare at 15', those labeled ~ are at 135'. Indicated on eachspectrum are the regions in which the angular distribu-tion was symmetric or isotropic.

0 I 2 5 4 5 6 7 8 9U (MeV)

FIG. 3. Same as Fig. 2 for the ~ Nb(P, n) Mo reaction.

Page 4: ) Cross Sections of Y, Nb, and Ta

610 GRIMES, ANDERSON, POHL, McCLURE AND WONG

"Nb(p, n), and '"Ta(p, n} reactions, respectively.At low energies (E~ ~ 10 MeV) the continuum spec-tra were either symmetric or isotropic, while forhigher energies a forward-peaked angular distri-bution was observed.

Backward angle (135') spectra for a number ofdifferent bombarding energies are shown in Figs.5-7. The lines drawn through the low-energyspectra for the Y(p, n) and Nb(p, n) reactions wereobtained by connecting the data points for the ap-propriate 10.0-MeV spectrum; as can be seen, thesame shape is also observed at lower energies. Itis concluded that the (P, n) reaction mechanism isessentially entirely compound nuclear in this ener-gy region for these two reactions. The Ta spectraare consistent with a completely compound-nu-

10

-I10

E =7.8 MeVX P

X

/X

E&=8.7 MeV~~

0

clear-reaction mechanism only below bombarding

energies of 9.6 MeV.In each case a constant-temperature level-den-

sity form is seen to be appropriate for excitationenergies between 3 MeV and the neutron binding

energy. The nuclear temperatures obtained were0.70, 0.68, and 0.54 MeV for ' Zr, Mo, and ' 'W,

respectively. Below 3 MeV, the level densitiesshow too much structure to be characterized with

a temperature, although the average slope is notsignificantly different from that above 3 MeV.

At higher energies the continuum spectra werefit with an expression of the form

-I10

-210

10

X X X

XX X

X

t' ~

X ~

10

X

cf

jJJ~IO-b

Fp=IO. O M

Xx

XXXX ~

xxX

~ ~XX ~

X

~ ISOT ROP I CX

-6IO ~ ~ 0

n ~ XX

XX ~

xxx,~ ~

ISOTROPIC

E =l4. 8 MeV ~

P'

~~ ISOT ROPIC

x"~XX ~

X ~xxxx ~

XXX~X

x ~ ~ ~XX XX ~ ~

XX

~ y

~ '~~ ~ ~ e$

~ ~~ ~ ~

-210

10

LLI

O

w 10CL

b

-510

-6IO

-710

X

}'o,0

I

0/

T=0.7 MeV —,0

0&0E =12.3 MeV

/00

/

,6

0/0

X

0 // X/

0 0/

0

XX

X

X/

X

X

Xx'

X/

Xg

x /

E =14.8 MeV

xXX

XX /

x

x

X

~E =100 Mev/x p

X-X~/

X

/

I'

Xx

/

-8 I gIG — E =8.0 MeV [xP

XX x

-8IO X

X

-9IO

X X ~

x xx X

r ~x

~ ~

-9I I I I I I I I I

0 I 2 5 4 5 6 7 8 9

U (MeV)

IOI I I I I I I I I I I

0 I 2 5 4 5 6 7 8 9 IO I I 12

U (MeV)

FIG. 4. Same as Fig. 2 for the isiTa(P, n)tsf% reaction,except that the points marked x are at 45'.

FIG. 5. Comparison of 135 neutron spectra producedin the ~Y(p,n) ~Zr reaction. The solid line was obtainedby connecting the data points in the 10.0-MeV spectrumand is extended to higher energies with the best-fit con-stant-temperature form.

Page 5: ) Cross Sections of Y, Nb, and Ta

EQUILIBRIUM AND NONEQUILIBRIUM CONTRIBUTIONS. . . 611

c(e, E)/E =Ap(U) +BU" . (7)

The nuclear temperatures determined at lower en-ergy were used in a constant-temperature level-density form for p(U) and the best-fit values of Aand I3 were calculated for 0 &n &6. Figure 8shows two examples of the quality of fit obtainedfor the best-fit value of n, and Fig. 9 displays thechanges in fit as a function of n. As can be seen,the best-fit value of n could be determined to with-in a range of 1 or 2. The "'Ta(P, n)' e'W data at14.8 MeV showed essentially no equilibrium con-tribution but in all other cases nonzero values forA were obtained.

To test the uniqueness of the fits, an alternativeform was tried

where 0 & n & 3, n+1 &m & 6. This parametrizationcan represent the spectrum over a limited range,but the fits showed systematic deviations from thedata for high U. In no case did Eq. (8) provide abetter fit than Eq. (7); for the 14.8-MeV Ta(p, n)data, where as already mentioned a very smallvalue for A was obtained, the fits were equallygood.

The best-fit values of n obtained with Eq. (7)are listed in Table II. A value of 1' would be ex-pected for either a direct or a pre-equilibrium re-action mechanism if the single-particle level spac-ing is independent of energy and if higher-order

10

v(e, E)/E =BU "+CU,

IOE =78 MeV

/

(8)

-I10

/0/

0/

/

-I10

-210

-3lo

LLJ)10

b

-510

-610

-7IO

-810

XX

X

00

0 0

0I

0 EP=8.7 MeV0/0

X p/

/0/0

/0/

pp0

0

00

EP=12.3 MeV /'0

,00

00

0, T=0.68 MeV

0 X

II

/X

„/ EP=IO.O MeV/

X

X/

XX'

X

/X

X/

X

"-XXX

/X

10

-310

//e

/ ~

E =8.0 MeV/P ~ ~

/

~0 000

/0

00

X/

X

/X

/X

/X

XX

/X

/X

X

/XX/

X X

10 /E =8.7 MeV/

P

0

/

I

//0

/

/

/

/

LLJ

~ 10«z -5LIJ

xx/x/

x/

X~

MeV/

X

/x

b

10—-6 E =9.6

-710—/

X

XX

x

xx /xx /XX

xx /

10

-9 X

EP=12.3 MeV xP

/~). T=05O Mev

~//

~ ~~ '/

E =Io.o Mev /

IO

-II I I I I I I I I

0 I 2 3 4 5 6 7 8 9U(MeV)

FIG. 6. Same as Fig. 5 for the 3Nb(P, n)~ Mo reaction.

1000

/I I L I I I

Z S ~ 5 6 7U (Mev)

FIG. 7. Same as Fig. 5 for the Ta(p, n) W reaction,except that the dashed line is the best-fit constant-tem-perature form obtained at low bombarding energies.

Page 6: ) Cross Sections of Y, Nb, and Ta

GRIMES, ANDERSON, POHL, McCLURE, AND KONG

F =14.8 MeV

10

=15

"j lo

JLLjCL lo

terms in U can be ignored. In spite of this pre-diction, the values 2, 3, and 5 are obtained for the"Y(P n)"Zr "'Ta(P n)"'W, and 9'Nb(P n) 'Mo

reactions, respectively. The fits were originallycarried out in the energy region between U= 2

MeV and the smaller of the two quantities: theneutron binding energy and the highest energy forwhich data were available. It was subsequentlyfound that reducing the limits to a region as smallas 4 ~ U & 6 MeV did not change the best-fit valuesof n obtained, although it reduced the differencebetween the best and worst fits. Thus, the mag-nitudes obtained for n are not due to a local anom-aly, but are characteristic of excitation energies

below the binding energy for these nuclei. Possi-ble explanations for the large n values will be pre-sented in Sec. IV. From these fits the dependenceof A and B on angle and bombarding energy werealso obtained.

If the term Ae ' represents the equilibrium con-tribution, an isotropic or symmetric" angular de-pendence for A(II) would be expected. Since B(B)may in general contain contributions from both di-rect and pre-equilibrium processes, a forward-peaked angular distribution for B(0) would be pre-dicted.

As can be seen from Fig. 10, these predictionswere verified. Also shown for comparison in Fig.10 is the 14.8-MeV Y(p, n) angular distribution forneutrons leaving ' Zr in the ground and first ex-cited states. Since these residual states shouldhave more overlap with the "Y ground state thanwould be expected for higher excited states, it isreasonable to assume that the angular distributionto these states should be more characteristic ofdirect reactions than would the nonequilibriumcontributions to continuum states. These two tran-

E =14.8 MeV

IO

-3IO

IO I I I I I I I I I

0 I 2 5 4 5 6 7 8 9IO

IO

U (MeV)

8=I&r0

LLI

4J~ IO

W IPo

Ld~ lo

b

b

IO

IO~ ~

I I t I I I I I I

p I 2 3 4 5 6 7 8 9

IO2 5 6 7

U (MeV)U(Mev}

FIG. 8. Fits of the form Ae ~ +BU" to the 14.8-MeVY(p,n) ~Zr neutron spectrum. Both a forward and a

backward angle are shown. The experimental spectrumis denoted with ~, the Ae term contribution (equilibri-um) with +, the BU" contribution (nonequilibrium) with

x, and the sum Ae ++BU" with 0.

FIG. 9. Variation of the quality of fit to the neutronspectrum of the expression Ae +BU" as a function ofn. The 14.8-MeV Y(p,n) 9Zr spectrum at 135' is theexample shown. The experimental spectrum is denotedx, and the best-fit spectra for m =0, 2, and 4 are de-noted by the dot dash, the solid line, and the dashed line,respectively.

Page 7: ) Cross Sections of Y, Nb, and Ta

EQUI LIBRIUM AND NON EQUILIBRIUM CONTRNTRI BUTIONS. . . 613

sltlons are ~ to ~' and ~ to ~ for the ground

and first excited states, respectively; the com-bination of two such different transitions would

be expected to be more representative of the con-tinuum angular distribution for direct processesthan would either individually. Inspection of Fig10 lndlcates that adding an isotropic component tothe form obtained for the first two levels can re-produce roughly the shape observed for the non-equilibrium portion. With this decomposition oneestimates equal contributions to the nonequilibri-um cross section from direct and pre-equilibriumprocesses.

B. Level Densities Above the NeutronBinding Energy

where 0 E, E,' is the cross section for a com-pound-nuclear reaction initiated by particle o. with

energy E leading to the emission of particle P with

ener E crgy „~,b, E' is the absorption cross sectionfor particle a at an energy E, as;„, (E') is the ah-sorption cross section for particle P at an energenergy

, p(U) xs the level density of the residual nucle-us, and Q is the Q value for the reaction (n tI). Ifthee level density is assumed to have a constant-

7

temperature energy dependence and if o;„,(E') isassumed to be a constant, the following results:

f E &Ce(E+0 E')irdEI -p(0

Above the neutron binding energy the level den-sity cannot be obtained from the shape of the evap-oration spectrum directly, because of the pres-ence of (P, 2n) neutrons in the spectrum. An al-ternative technique proposed by Ericson, "can beused to extend the level-density measurementsbe ond thiy 's point ~ For a reaction which proceedsto a given final state through the compound nu-cleus, the energy dependence of the cross sectionis determined primarily by the level densities int e residual nuclei reached by particle emissionThe Coulomb barrier will inhibit competition fromnuc ei reached by charged-particle emission; thus,the energy dependence of the cross section forthese nuclei will be determined primarily b thy e

c e y neutronvel density of the nucleus reached b nemission.

The following relation is obtained b W'

kwing

y elss opf

4J

3-LdCL

F =l4.8 MeV

E E ) = loBinv (EI)(Ta abc(E)E+&f E'os;„,(E ')p(E+ Q E')dE'-

TABLE II. Best-fit n values.

(MeV) 150 45' 75' 105' 135'

12.314.S

10.012.314.0

10.411.712.314.8

3, 455

3, 4432

8Y(p, n) Zr

1, 2

1 2

93Nb(P, n) 83Mo

3 3, 45 5

5, 6 4, 5

18iTa(p n) 18i~3 33 33 2

3 3

1 22

44, 55

43, 433

3—555, 6

' o I I I ~ I ~ I ~ I

40 60 80 l00 I 20 140 l60

FIG. 1089' n "Zr

0. Dependence of A(E, 6) and B(E 0) on 0 for the(p,n) r reaction at 14.8 MeV. The A(E, 8) values

(equilibrium) are deare enoted+ and the estimated angular dis-tribution is denoted by the dashed linefrom the B I9

as e ine. Contributionsom e g, (9) term are labeled with an x and th

from the (p,n +nn an ose

,no+ n~) groups with a O. In each case thproximate a ar

case e ap-

line.ar distribution is indicated with 1 dar ' ' ' asoi

Page 8: ) Cross Sections of Y, Nb, and Ta

614 GRIMES, ANDERSON, POHL, McCLURE, AND WONG

Since the right side of Eq. (10) does not dependon F.„ the cross sections to a number of levels,divided in each case hy the outgoing energy, canbe summed to obtain

~ & 8(E~ Ei) &&~.~~ (E)&p(E+ 9) '

where A' is the number of levels included in theanalysis. The technique used in separating theequilibrium and nonequilibrium contributions tothe spectra yields a parametrization of the equi-librium portion in the form

[o(E,)/E, ]„„„=Ae"', (12)

p(E+0)~o. .„(E)/A(E) .

Thus, the energy dependence of the level densityfor energies above the neutron binding energy canbe obtained from the variation of A(E) with bom-barding energy.

The results obtained for "Zr, "Mo, and "'Ware shown in Figs. 11-13. Potential parametersproposed by Becchetti and Greenlees" were usedto calculate the proton absorption cross sections.

where A is a function of bombarding energy. Com-parison of Eqs. (11) and (12) shows that

Since only the relative values of o„,b, (E) are need-ed, the results are not particularly sensitive tothe potential parameters used. The optical-modelpotential of Hodgson ' yielded absorption crosssections which differed by 30-40%%u~ from those ofBecchetti and Greenlees, but because the energydependence is similar, use of the Hodgson crosssections would change the relative values of thelevel density by less than 10%. Each A(E) valueyields a relative measurement of the level densityat the excitation energy corresponding to the in-cident center-of-mass energy minus the Q valuefor the (p, n) reaction; thus, there is some over-lap between the region covered with this techniqueand that obtained directly from the spectral shape.In each case, the energy dependence extracted us-ing the two techniques agrees within errors. Theconstant-temperature level-density form is foundto be appropriate for excitation energies up to 10MeV. For the appropriate choice of a, the excita-tion-energy dependence of the usual Fermi-gasform (p(U) =[1/(U —5)2]e't'U ~'i ) does not differappreciably over an energy range of a few MeVfrom that of the constant-temperature form; ashas been pointed out by Maruyama, " the level den-sities for certain nuclei can be fit with eitherform over a limited energy range. The presentmeasurements indicate that the constant-temper-

10

10—

UJ)I—

10

//

//f

// T=0.7 MeV

IO

QJ

W P10—

/(/

//T=0. 68 MeV

/

I

w 10

/10

7 8 9U (Mev}

l I

10 I I 12

O 10 //

/

/

l )

6 7 8 9 10 II 12 13

U (MeV)

FIG. 11. Level density of Zr obtained from the bom-barding energy dependence of A(E, 8) . The dashed linerepresents the form obtained for energies below the neu-tron binding energy from the neutron spectral shape.

FIG. 12. Level density of ~3Mo obtained from the bom-barding energy dependence of A (E, 0). The dashed linerepresents the form obtained for energies below the neu-tron binding energy from the neutron spectral shape.

Page 9: ) Cross Sections of Y, Nb, and Ta

EQUILIBRIUM AND NONEQUILIBRIUM CONTRIBUTIONS. . . 615

UJOI—

p10

181

/T=O. 54 MeV

the residual nucleus, does not depend on either ofthese quantities, because of the proportionality of

p„,(U} and o (e}/e. Values for I';„, obtained fromthis relation are shown in Table III. The single-particle level spacing (g=6a/w') necessary for cal-culating the level densities was obtained from therelation a= SA, yielding values of 7 for Zr andMo and 14 for W'. The uncertainty in these widthsis substantial, primarily because of the difficultyin estimating the fraction of the nonequilibriumcross section due to direct processes. The esti-mate of equal direct and pre-equilibrium crosssection is uncertain by at least 5(@. As can beseen from Table III, the values for l;„,are approx-imately 300 keV for A near 90 and about 1MeVfor A near 180. An earlier determination' of I';„,yielded a value of 160+ 80 keV for A near 55.

)//

D. Relative Magnitude of Equilibriumand Nonequilibrium Cross Sections

The analysis described in Sec. IIIA provides aparametrization of the spectra in the form

o/e =AeU' +BU". (15)I

7 9 IO I 1

U (Mev)l2

FIG. 13. Level density of ~s~W obtained from the bom-barding energy dependence of A(E, ej. The dashed linerepresents the form obtained for energies below the neu-tron binding energy froIn the neutron spectral shape.

ature energy dependence is observed over a largerrange than is consistent with a Fermi-gas form,Changes in temperature are observed above 10MeV and it is felt that these represent the transi-tion to a Fermi-gas energy dependence at higherexcitation energies.

C. Intermediate-Level Widths

Application of the results of Williams' and Fesh-back, Kerman, and Lemmer' yields a relation be-tween the pre-equilibrium cross section and thewidth of the pre-equilibrium state from which de-cay has occurred.

Solving Eq. (5) for I';„„ the following result isobtained:

Because of the linear bias on the detector, neu-trons with energies less than 1.6 MeV were notdetected in this experiment. Thus, total crosssections for neutron production cannot be extract-ed directly from the data.

If the parametrization expressed in Eg. (15) isassumed to be valid over the entire energy rangeof the neutron spectrum, the integral of the ex-pression will yield the neutron-production crosssection, including those parts due to equilibriumand nonequilibrium processes. For higher bom-barding energies, all of the neutrons correspond-ing to (p, n) events will have energies above the de-tector bias and thus the (p, n) cross sections canbe obtained directly. For these cases, the inte-gral was extended beyond the neutron binding en-ergy to the maximum possible excitation energy(zero neutron energy), to provide a value for thesum of the (p, n) and (p, 2n) cross sections. If, asis expected for these nuclei, cross sections forcharged-particle emission from the compound nu-

TABLE III. Intermediate-resonance widths.(2S+ l)o,~, mo„„p„,(U) (14)v' hip„(E ~)[o~,(e)/~ jwhere o,b, is the absorption cross section in theentrance channel, O~p is the inverse capturecross section (exit channel), m is the mass of theemitted particle, E* is the excitation energy in thecompound system, and p„(E*)is the density of nexciton states at an energy 8*. The above expres-sion, although it contains c, the energy of theemitted particle, and U, the excitation energy of

Nucleus

iS2~

Compound-nucleusexcitation

(NeV)

20.52320.822.519.121.6

Intermediate-resonance width

(keV)

240+120320 + 160340+170660+330610 ~305

1400 ~ 700

Page 10: ) Cross Sections of Y, Nb, and Ta

616 GRIMESs ANDERSON~ KOHL~ McCLURE& AND WONG

cleus are small, the above sum should be approx-imately equal to the reaction cross section. Thiscomparison provides an additional approximatetest of the appropriateness of the parametrizationof Eq. (15), although possible discrepancies couldresult from changes in the parameters of the lev-el-density formula for high U or variations in theinverse capture cross section for low neutron en-ergies.

The results of these calculations are shown inTable IV. In all cases, the sum of (p, n} and

(p, 2n} cross sections is approximately equal tothe reaction cross section calculated with the Bec-chetti parameters. The integral of the AeU'~ termyields more cross section than the integral of theBU" term, a result consistent with the conclusionthat the damping width of the doorway state intomore complicated states is much larger than thatto the continuum. If the limit of integration ischosen to be the neutron binding energy, approxi-mate values for the equilibrium and pre-equilib-rium contributions to the (p, n) cross section canbe obtained; as can be seen, essentially all of the'~'Ta(p, n)'"W cross section is due to nonequilib-rium processes at 14.7 MeV, as are substantialportions of the ' Y(p, n)89Zr and "Nb(p, n)"Mocross sections. These calculations depend on theassumption that a second neutron will be emittedwhere this is energetically possible; because of

competition from y decay, this represents only a

lower limit for the (p, n) cross section. Table I~''

also presents values of the ratio I' „/I';„,, the(neutron) width to the continuum divided by the to-tal width of the intermediate states. Since to firstorder the incoming proton will produce a neutronparticle-hole state as often as a proton particle-hole state, on the average three times as manyprotons as neutrons will be in excited states. TheCoulomb barrier will inhibit proton emission rela-tive to neutron emission; nonetheless, it is ex-pected that the proton width would be somewhatlarger than the neutron width.

V. DISCUSSION

A. Level Densities

Previous determinations of the level-density pa-rameters of nuclei studied in the present workare listed in Table V. The level density of "Mohas bpen studied by Bramblett and Bonner" and

by Wong et al." Comparison of the Bramblett andBonner results with the present measurements isdif6cult because the former study is restricted toexcitation energies below 4 MeV, while the pres-ent measurements indicate that for excitation en-ergies less than 3 MeV considerable structure ispresent in the level density. They also find thata constant-temperature form fits the data, but ob-

TABLE IV. Equilibrium and nonequilibrium neutron cross sections.

TargetEp

(MeV) cr(p, n) cr(P, n+2n)

Equilibriumcross section

(mb)cr(P, n) o(P, n+ 2n)

Nonequilibriumcross section

(mb)

Con~ int

89'

181Ta

"v'

9coa

10.012.314.8

8.710.012.314.0

8.710.010.411.712.314.8

12.313.214.7

12.313.214.7

800 +1501250 + 200

540 +150

760 +150800 +200

75 +2516 +5

77+ 2036 +1030 ~106+22~1

750 + 100460+100180 + 50

520 + 100320+ 100100 + 40

1620 +400

1100+ 3501000 ~ 3501700 + 800

110+30330 + 140580 ~ 300900+400800+400

1040 + 200990+ 300925 ~ 350

810 + 200740+ 350820+ 350

67+2088 +30

66+1575+2536+15

6+39+5

15 +515 ~520 +8

40 +2050 +2090 +40

42 +2030 +1540 +15

100 +40

70 ~30200 +80170 +80

9+515+1037 +1543+1570+30

43~2564 +40

120 +60

46+2533+2063 +30

0.03 ~0.010.03 +0.01

0.03 + 0.010.1 +0.030.05 + 0.04

0.015 + 0.010.015 + 0.010.02 ~0.010.03 + 0.01

0.02 + 0.010.03 + 0.010.07 + 0.03

0.03 + 0.010.02 x 0.010.04 + 0.02

Page 11: ) Cross Sections of Y, Nb, and Ta

EQUILIBRIUM AND NONEQUILIBRIUM CONTRIBUTIONS. . .

tain a temperature of 0.57 MeV. The results ofWong et aI,."indicate that the level density of "Mohas a Fermi-gas form between excitation energiesof 3 and 8 MeV; the parameters they obtain forthe fit result in a variation of the nuclear temper-ature between 0.7 and 0.86 over this range, in ex-cellent agreement with the 15' "Mo temperaturesquoted in Table I. It is now evident that the Fermi-gas dependence observed in Ref. 18 for "Mo wasdue to neglect of nonequilibrium contributions.

The level density of "'W has also been measuredby Holbrow and Barschall, "using the "'Ta(P, n)-"'W reaction. They find that the shape of the evap-oration spectrum changes quite rapidly with pro-ton energy and obtain temperatures of 0.64, 0.70,and 0.78 MeV at bombarding energies of 8, 9, and10 MeV, respectively. This trend is consistentwith that observed in the present measurements(see Table I) and is attributed to the presence ofnonequilibrium processes. These temperaturesare systematically somewhat higher than those ob-tained in the present measurements. The explana-tion for this discrepancy is not clear, although thesmall cross sections for the (P, n) reaction on Tamake the results more sensitive to the backgroundsubtraction than is the case for the other elementsstudied.

The level densities for all three nuclei measuredin the present study have been obtained previouslyby Borchers, Wood, and Holbrow. " Again, their

results could not be reconciled with the tradition-al equilibrium assumption, because of a depen-dence of the level density on bombarding energyas well as residual excitation. The values for thenuclear temperature obtained at the lowest bom-barding energies are in reasonably good agree-ment with the present results.

Verbinski and Burris" have measured the (P, n

+ 2n) spectra produced by 18-MeV proton bombard-ment of Ta. They quote a value of 1.04 MeV forthe average temperature of ' W and ' W. Theequation derived by LeCouteur and Lang" wasused to obtain this temperature from the 2n por-tion of the spectrum; as will be discussed later,there is some indication that the 2n spectrum isnot accurately described by this relation, whichmight account for the discrepancy between thetemperature obtained by Verbinski and Burris andthat determined in the present measurement.

Level densities for neighboring nuclei have beenmeasured by Maruyama, "Owens and Towle, "Thomson, ' Plattner et al. ,

"Sobottka et al. ,"andMathur et al." The level-density forms obtainedby Maruyama for Y and Nb with the (n, n') reac-tion could be fit with either a Fermi-gas or con-stant-temperature form; the values for the tem-perature were about 0.7 MeV, in excellent agree-ment with the present measurements. The sametechnique was used by Thomson to obtain level-density parameters for Nb, Ta, and W. He ob-

TAB LE V. Level-density parameters.

Nucleus Reference MethodBombarding

energy U rangea

(Me V-')T

(MeV)

89'

"Zr

93Mo

Mo

si Ta

1627

20Present work

1624

171820

Present work

2527

162326

192021

Present work

(n n'){n,n' + 2n)

(p, n)

(p, n)

(n, n'){n,n')

(p, n)

(p, n)

(p, n)

(p, n)

(n, n' + 2n)

{n,n' + 2n)

(n n')(n, n'}{n,n' + 2n)

(p, n)

(p, n)

(p, n +2n)(p, n)

(n, n')(n, n' +2n)

2 —7.514

9—127.8 —14.8

2 —7.54-7.05.37—139-127.8—14

1414

2 —7.55—714

8 —109—12188—14.8

5—714

2 —7,5

2 —82 —9

2 —7.5

c43-82 —82 —8

2-7.53—7

2 7

3-7

13.5—21.713.3

14.6-22.722.5—38.68.811.411

13.510.221.5-26.716.4 —17.723-27

25

16.0—16.621.7

0.63-0.840.93

0.7-1.00.7

0.6-0.780.33—0.59

0.570.7-0.860.7-1.050.68

0.901.02

0.56-0.630.82

0.64-0.780.5-1.051.040.54

0.52-0.60.72

Page 12: ) Cross Sections of Y, Nb, and Ta

GRIMES, ANDERSON, POHL, McCLURE, AND WONG

tained temperatures of 0.59, 0.52, and 0.5 MeV,respectively. In the case of Nb, however, the nu-clear temperature was found to depend on born-barding energy, indicating that some nonequilib-rium processes might have been present. Similarresults have been reported by Owens and Towle,who obtain temperatures of about 0.55 MeV forboth Ta and W, but find a dependence of temper-ature on bombarding energy as well. Level-den-sity parameters for Mo and Ta have been obtained

by Plattner et al. and Sobottka et al. , respectively,and those for Y, Mo, and W by Mathur et al. Thesedata were obtained from 14-MeV neutron bombard-ment and extraction of the level-density parame-ters is complicated by the presence of (n, 2n) neu-trons. Temperatures of 0.9 for Mo and 0.82 forTa were obtained in the first two experiments andvalues of 0.93, 1.02, and 0.72 for Y, Mo, and W,respectively, were obtained by Mathur et al.There is some indication" that use of the Langand LeCouteur formula" yields level-density pa-rameters for Ta which do not agree with those ob-tained by calculating the spectral shape directly;this might imply that the Lang and LeCouteur re-lation, which was derived for neutron cascades,is not an accurate representation of an (n, 2n) spec-trum.

B. Reaction Mechanisms

The present measurements indicate that non-equilibrium processes are more easily observedfor nuclei with A near 90 and 180 than for nucleinear A= 55. The lower temperatures characteristicof these nuclei produce a more rapid decrease inthe equilibrium cross section to given final statesas the bombarding energy is increased, with theresult that nonequilibrium contributions can be ob-served more easily. Comparison of the presentresults with those of Ref. 1 indicates that the mag-nitude of the nonequilibrium contributions to the(P, n) cross section varies only slightly betweenselected nuclei in the A = 50, A = 90, and A = 180regions. Thus, the ratio I', (I';„„where I' „ isthe continuum width and I';„, the total width of thepre-equilibrium state, does not change significant-ly for the nuclei investigated. The total width Ihowever, does increase from about 160 keV toabout 1 MeV between A = 55 and A = 181. Theoret-ically, this width is expected to be proportional tog'8*', where g is the single-particle level spac-ing and E* is the excitation energy. If g is as-sumed to be proportional to A, this would resultin a value of over 30 for the ratio I';„,(A = 181)/I;„,(A = 55), while the experimental value is 8.

Application of the level-density formulas ob-tained from the assumption of a constant single-particle level spacing leads to the prediction that

the nonequilibrium spectrum will have a linear de-pendence on U for both the direct and pre-equilib-rium portions of the (p, n) spectrum. Disagree-ment with this prediction was observed for the re-actions "V(p, n)"Cr and ' Co(P, n)' Ni, for which aquadratic dependence on U produced the best fit tothe data. This discrepancy was not regarded asserious, because of (1) the uncertainty of one unitin the n determination procedure, and (2) smallcontributions from higher-order terms in the ser-ies. The present measurements especially Nb(P, n)show much larger disagreements with theoreticalpredictions and are too large to be explained bythe above factors. A more probable explanation isthat because of shell effects, the single-particlelevel spacing is not sufficiently independent of ex-citation energy to be regarded as constant. Thebasic assumption of the model is that the relevantresidual states are one-particle (proton)-one-hole(neutron) states. Such a level density would be ex-pected to show the effects of non-uniform single-particle level spacing more dramatically thanwould the total level density which includes all con-figurations. Since the effect of shells is to pro-duce large gaps between ground or low-lying statesand higher excited states, the result is to reducethe rate of increase in level density for small val-ues of U relative to that for higher U; this effectof a gap would be in the proper direction to causethe "best-fit" value of n to be larger than it wouldbe in the absence of shell effects. Similar behav-ior in the neighborhood of closed shells has beennoted by Cline and Blann. " Calculations by Maru-yama" also have shown that shell effects would bemore obvious in the constituent level densities(e.g. proton particle states) than in the compositelevel densities. It is clear that measurements fora number of nuclei near closed shells will be nec-essary to determine whether this behavior is char-acteristic of nuclei near closed shells or unique tothe nuclei studied here.

VI. SUMMARY

Examination of the neutron spectra from (p, n)reactions on 8'Y 9sNb and &8xTa has shown thatsubstantial contributions from nonequilibrium pro-cesses are present for bombarding energies above10 MeV. Data obtained at lower energies wereused to obtain the level-density parameters for

Zr, Mo, and '"W; a constant-temperatureform is found to be appropriate for energies belowthe neutron binding energy, with temperatures of0.7, 0.68, and 0.54 MeV, respectively.

Spectra at higher bombarding energies were re-solved into equilibrium and nonequilibrium por-tions. Use of the bombarding-energy dependence

Page 13: ) Cross Sections of Y, Nb, and Ta

EQUILIBRIUM AND NONEQUILIBRIUM CONTRIBUTIONS. . . 619

of the compound-nuclear portion enabled the level-density measurements to be extended to about 12MeV. Changes in temperature consistent with theexpected transition to a Fermi-gas level-densityform were observed above 10 MeV.

The characteristics of the nonequilibrium partof the spectrum were found to be in basic agree-ment with the predictions of the current model.Inconsistencies in the residual excitation-energydependence of the spectra were observed and ten-

tatively attributed to shel1. effects.Comparison of the magnitudes of the equilibrium

and nonequilibrium cross section provided an esti-mate of the intermediate-state width and that frac-tion of the width corresponding to particle emis-sion. The widths varied between approximately300 keV for "Zr and "Mo to 1 MeV for '"W; thewidth for nonequilibrium particle emission wasfound to be between 5 and 10%%up of the total width otthese states.

)Work performed under the auspices of the U. S. Atom-ic Energy Commission.

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~V. A. Sidorov, Nucl. Phys. 35, 253 (1962).'R. N. Wood, R. R. Borchers, and H. H. Barschall,

Nucl. Phys. 71, 52S (1965).H. Feshbach, A. K. Kerman, and R. H. Lemmer, Ann.

Phys. (N.Y.) 41, 230 (1967).The factor (2$+1) was inadvertently omitted from the

corresponding equation in Ref. 1.J. D. Anderson and C. Wong, Nucl. Instr. Methods 15,

178 (1962); B. D. Walker, J. D. Anderson, J.W. McClure,and C. Wong, Rid. 29, 333 (1964).

~~T. Ericson, and V. Strutinski, Nucl. Phys. 8, 284(1958).

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(1940).4F. D. Becchetti and G. W. Greenlees, Phys. Rev. 182,

1190 (1969).~SP. E. Hodgson, in Proceedings of the International

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Inc. , New York, 1963), p. 103.M. Maruyama, Nucl. Phys. A131, 145 (1969).

'R. L. Bramblett and T. W. Bonner, Nucl. Phys. 20,395 (1960).

C. Wong, J. D. Anderson, J. W. McClure, and B. D.Walker, Nucl. Phys. 57, 515 (1964).

~9C. H. Holbrow and H. H. Barschall, Nucl. Phys. 42,264 (1963),

~OR. R. Borchers, R. N. Wood, and C. H. Holbrow,Nucl. Phys. 88, 689 (1966).

V. V. Verbinski and W. R. Burrus, Phys. Rev. 177,1671 (1969).

K. J. LeCouteur and D. W. Lang, Nucl. Phys. 13, 32(1959).

R. O. Owens and J. H. Towle, Nucl. Phys. A112, 337(1968).

2 D. B. Thompson, Phys. Rev. 129, 1649 (1963).~R. Plattner, P. Huber, C. Poppelbaum, and R. Wag-

ner, Helv. Phys. Acta 36, 1059 (1963); P. Huber,R. Plattner, C. Poppelbaum, and R. Wagner, Phys. Let-ters 5, 202 (1963).

~H. Sobottka, S. Grimes, P. Huber, E. Mangold,J. Schacher, and R. Wagner, Helv. Phys. Acta 43, 559(1970).27S. C. Mathur, P. S. Buchanan, and I. L. Morgan,

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